The present invention relates generally to synchronizing chaotic systems and more particularly a system which allows the synchronizing of one chaotic system to another chaotic system using only a narrow band signal.
A synchronized nonlinear system can be used as an information transfer system. The transmitter, responsive to an information signal, produces a drive signal for transmission to the receiver. An error detector compares the drive signal and the output signal produced by the receiver to produce an error signal indicative of the information contained in the information signal.
It is known to those skilled in the art that a nonlinear dynamical system can be driven (the response) with a signal from another nonlinear dynamical system (the drive). With such a configuration the response system actually consist of duplicates of subsystems of the drive system, which are cascaded and the drive signal, or signals, come from parts of the drive system that are included in the response system. FIG. 1 shows a cascaded chaotic system 100 known in the prior art. Drive system 100 comprises a chaotic drive circuit 140, housed in transmitter system 139, and a chaotic response circuit 160, housed in a receiver system 166. Chaotic drive circuit comprises subsystems 198 and 199 which are duplicated by subsystem 169 and 170 in the response circuit. A nonlinear function 150 is contained in drive circuit 140 and is used to drive the system into chaotic operation.
A chaotic system has extreme sensitivity to initial conditions. The same chaotic system started at infinitesimally different initial conditions may reach significantly different states after a period of time. Lyapunov exponents (also known in the art as xe2x80x9ccharacteristic exponentsxe2x80x9d) measure this divergence. A system will have a complete set of Lyapunov exponents, each of which is the average rate of convergence (if negative) or divergence (if positive) of nearby orbits in phase space as expressed in terms of appropriate variables and components.
Sub or Conditional Lyapunov exponents are characteristic exponents which depend on the signal driving the system. It is also known to those skilled in the art that, if the sub-Lyapunov, or conditional Lyapunov, exponents for the driven response system are all negative, then all signals in the response system will converge over time or synchronize with the corresponding signals in the drive. When the response system is driven with the proper signal from the drive system, the output of the response system is identical to the input signal. When driven with any other signal, the output from the response is different from the input signal.
In brief, a dynamical system can be described by the equation
dxcex1/dt=f(xcex1).xe2x80x83xe2x80x83(1)
The system is then divided into two subsystems. xcex1=(xcex2,"khgr");
dxcex2/dt=g(xcex2, "khgr")
d"khgr"/dt=h(xcex2, "khgr")xe2x80x83xe2x80x83(2)
where xcex2=(xcex11 . . . xcex1n), g=(f1(xcex1) . . . fn(xcex1)), h=(fn+1(xcex1) . . . fm(xcex1)), "khgr"=(xcex1n+1, . . . xcex1m), where xcex1, xcex2 and "khgr" are measurable parameters of a system, for example vectors representing a electromagnetic wave.
The division is arbitrary since the reordering of the xcex1i variables before assigning them to xcex2, "khgr" g and h is allowed. A first response system is created by duplicating a new subsystem "khgr"xe2x80x2 identical to the "khgr" system, and substituting the set of variables xcex2 for the corresponding xcex2xe2x80x2 in the function h, and augmenting Eqs. (2) with this new system, giving,
dxcex2/dt=g(xcex2, "khgr"),
d"khgr"/dt=h(xcex2, "khgr")xe2x80x83xe2x80x83(3)
d"khgr"xe2x80x2/dt=h(xcex2, "khgr"xe2x80x2).
If all the sub-Lyapunov exponents of the "khgr"xe2x80x2 system (i.e. as it is driven) are less than zero, then ["khgr"xe2x80x2xe2x88x92"khgr"]xe2x86x920 as t infinity. The variable xcex2 is known as the driving signal.
One may also reproduce the xcex2 subsystem and drive it with the "khgr"xe2x80x2 variable, giving
d"khgr"/dt=g(xcex2, "khgr"),
d"khgr"/dt=h(xcex2, "khgr"),
d"khgr"xe2x80x2/dt=h(xcex2, "khgr"xe2x80x2).xe2x80x83xe2x80x83(4)
d"khgr"xe2x80x2/dtxe2x80x2=g(xcex2xe2x80x3, "khgr"xe2x80x2)
The functions h and g may contain some of the same variables. If all the sub-Lyapunov exponents of the "khgr"xe2x80x2, xcex2xe2x80x3 subsystem (i.e. as it is driven) are less than 0, then xcex2xe2x80x3xe2x86x92xcex2 as txe2x86x92infinity. The example of the eqs. (4) is referred to as cascaded synchronization. Synchronization is confirmed by comparing the driving signal xcex2 with the signal xcex2xe2x80x3.
Generally, since the response system is nonlinear, it will only synchronize to a drive signal with the proper amplitude. If the response system is at some remote location with respect to the drive system, the drive signal will probably be subjected to some unknown attenuation. This attenuation can be problematic to system synchronization.
It is also known by those skilled in the art, that it is possible to pass chaotic signals from a drive system through some linear or nonlinear function and use the signals from the response system to invert that function as discussed, for example, in Carroll, et al., xe2x80x9cTransforming Signals with Chaotic Synchronization,xe2x80x9d Phys. Rev. E. Vol. 54, p. 4676 (1996).
The present invention builds on the design of three previous inventions, the synchronizing of chaotic systems, U.S. Pat. No. 5,245,660, the cascading of synchronized chaotic systems, U.S. Pat. No. 5,379,346, and a method for synchronizing nonlinear systems using a filtered signal, U.S. Pat. No. 5,655,022 each herein incorporated by reference. The present invention extends those principles to allow the synchronization of a broad band chaotic receiver to a broad band chaotic transmitter, using only a narrow band chaotic signal.
It is therefore an object of the invention to provide systems for producing synchronized signals, and particularly nonlinear dynamical (chaotic) systems.
Another object of the invention is to provide a chaotic communications system for encryption utilizing synchronized nonlinear transmitting and receiving circuits using a narrow-band version of the chaotic signal to synchronize the broader band chaotic transmitter and receiver.
A further object of the invention is to provide a chaotic communication system which employs a narrow band version of the chaotic signal for synchronizing transmitter and receiver units to facilitate efficient use on existing telephone or FM radio channels.
The present invention is an autonomous system design featuring subsystems which are nonlinear and possibly chaotic, but will still synchronize when the drive signal is attenuated or amplified by an unknown amount. The system uses filters to produce a narrow band version of the wideband chaotic signal to synchronize the chaotic transmitter to the chaotic receiver. The small bandwidth affords the system a greater resistance to the effects of noise, specifically the systems resistance to channel distortion and accompanying phase shifts is greatly increased by employing a narrow band. The broad band chaotic signal is transmitted and appears to be noise to an unauthorized listener. The receiving unit employs band pass filtering, and when the signal is received the receiver filters the broadband chaotic signal. The filters produce a narrowband chaotic signal and that narrowband signal is used to synchronize the transmitter and receiver.